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Probability

5.5 Convolution

Joshua Fan

Alex Tsun

​

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Agenda

  • Law of Total Probability for RVs
  • Sums of RVs by Convolution

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Law of Total Probability for RVs

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Law of Total Probability for RVs

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Law of Total Probability for RVs

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Law of Total Probability for RVs

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Convolution (Motivation)

Definition of

Expectation

Definition of

Expectation

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Convolution-Discrete (Idea)

X

Y

Z=X + Y

1

5

6

2

4

6

3

3

6

4

2

6

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Convolution-Discrete (Idea)

X

Y

Z=X + Y

1

5

6

2

4

6

3

3

6

4

2

6

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Convolution-Discrete (Idea)

X

Y

Z=X + Y

1

5

6

2

4

6

3

3

6

4

2

6

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Convolution-Discrete (Idea)

X

Y

Z=X + Y

1

​

6

2

​

6

3

​

6

4

​

6

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Convolution-Discrete (Idea)

X

Y

Z=X + Y

1

5

6

2

4

6

3

3

6

4

2

6

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Convolution-Discrete (Idea)

X

Y

Z=X + Y

1

5

6

2

4

6

3

3

6

4

2

6

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Convolution-Discrete (Idea)

X

Y

Z=X + Y

1

5

6

2

4

6

3

3

6

4

2

6

X, Y are independent!

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Convolution-Discrete (Idea)

X

Y

Z=X + Y

1

5

6

2

4

6

3

3

6

4

2

6

X, Y are independent!

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Convolution-Continuous (Idea)

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Convolution-Continuous (Idea)

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Convolution-Continuous (Idea)

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Convolution-Continuous (Idea)

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Convolution-Continuous (Idea)

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Convolution-Continuous (Idea)

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Convolution-Continuous (Idea)

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Convolution-Continuous (Idea)

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Convolution-Continuous (Idea)

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Summary: Convolution

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Random Picture

probability students

Definition of

Expectation

The “Normal” Distribution

The “Gaussian” Distribution

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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Convolution-Continuous (Example)

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END PIC

​

Alex Tsun

Joshua Fan